Modal Interpretations of Quantum Mechanics

The original “modal interpretation” of non-relativistic
quantum theory was born in the early 1970s, and at that time the
phrase referred to a single interpretation. The phrase now encompasses
a class of interpretations, and is better taken to refer to a
general approach to the interpretation of quantum theory. We shall
describe the history of modal interpretations, how the phrase has come
to be used in this way, and the general program of (at least some of)
those who advocate this approach.

1. The origin of the modal approach

In traditional approaches to quantum measurement theory a central role
is played by the projection postulate, which asserts that upon
measurement of a physical system its state will be projected
(“collapses”) onto a state corresponding to the value
found in the measurement. However, this postulate leads to many
difficulties: What causes this discontinuous change in the physical
state of a system? What exactly is a “measurement” as
opposed to an ordinary physical interaction? The postulate is
especially worrying when applied to entangled compound systems whose
components are well-separated in space. For example, in the
Einstein-Podolsky-Rosen (EPR) experiment there are strict correlations
between two systems that have interacted in the past, in spite of the
fact that the correlated quantities are not sharply defined in the
individual systems. The projection postulate in this case implies that
the collapse resulting from a measurement on one of the systems
instantaneously defines a sharp property in the distant other system.
(See the discussion of the collapse or projection postulate in the
entry on
philosophical issues in quantum theory.)

A possible way clear of these problems was noticed by van Fraassen
(1972, 1974, 1991), who proposed to eliminate the projection postulate
from the theory. Others had made this proposal before, as Bohm (1952)
in his theory (itself preceded by de Broglie’s proposals from
the 1920s), Everett (1957) in his relative-state interpretation and De
Witt (1970) with the many-worlds interpretation. (See the entries on
Bohmian mechanics,
Everett’s relative-state formulation of quantum mechanics, and the
many-worlds interpretation of quantum mechanics.)
Van Fraassen’s proposal was, however, different from these
other approaches. It relied, in particular, on a distinction between
what he called the “dynamical state” and the “value
state” of a system at any instant:

The dynamical state determines what may be the
case: which physical properties the system may possess, and which
properties the system may have at later times.

The value state represents what actually is the
case, that is, all the system’s physical properties that are
sharply defined at the instant in question.

The dynamical state is just the quantum state of the ordinary textbook
approach (a vector or density matrix in Hilbert space). For an
isolated system, it always evolves according to the
Schrödinger equation (in non-relativistic quantum mechanics): so
the dynamical state never collapses during its evolution.

The value state is (typically) different from the dynamical state. The
general idea of this original proposal, and of modal interpretations
in general, is that physical systems at all times possess a number of
well-defined physical properties, i.e., definite values of physical
quantities; these properties can be represented by the system’s
value state. Which physical quantities are sharply defined, and which
values they take, may change in time. Empirical adequacy of course
requires that the dynamical state generate the correct Born
frequencies of observable quantities.

An essential feature of this approach is that a system may have a
sharp value of an observable even if the dynamical state is not an
eigenstate of that same observable. The proposal thus violates the
so-called “eigenstate-eigenvalue link”, which says that a
system can only have a sharp value of an observable (namely, one of
its eigenvalues) if its quantum state is the corresponding eigenstate.
In the value state terminology, the eigenstate-eigenvalue link would
say that a system has the value state corresponding to a given
eigenvalue of a given observable if and only if its dynamical state is
an eigenstate of the observable corresponding to that eigenvalue. This
original modal approach accepts the “if” part, but denies
the “only if” part.

What are the possible “value states” for a given system at
a given time? Van Fraassen stipulates the following restriction:
propositions about a physical system cannot be jointly true, unless
they are represented by commuting observables. In other words, the
non-commutativity of observables imposes limits not on our
knowledge about the properties of a system, but rather on the
possibility of joint existence of properties, independently
of our knowledge. Non-commuting quantities, like position and
momentum, cannot jointly be well-defined quantities of a physical
system.

Empirical adequacy requires that, in cases of measurement, the
after-measurement value state of the apparatus corresponds to the
(definite) measurement result. Therefore, in these cases one would
expect the dynamical state to generate a probability measure over
exactly the set of possible measurement results. However, van
Fraassen’s original modal approach is more liberal in its
assignment of possible value states, and according to many this does
not yield a satisfactory account of measurements (see Ruetsche
1996).

Van Fraassen’s proposal is “modal” because it
naturally connects to a modal logic of quantum propositions. Indeed,
the dynamical state in general only tells us what is
possible. According to van Fraassen, one does not need to
view this as arising from an incompleteness of the description, which
it is the aim of science to remove—quantum mechanics may be
inherently probabilistic and modal (see Bueno 2014 for the relation
between this and van Fraassen’s constructive empiricism, which is
hostile to modal realism).

It is easy to see how, along the lines of van Fraassen’s ideas,
a program could come into being for providing a more elaborate
“realist” interpretation of quantum theory, a program to
which we now turn.

2. General features of modal interpretations

In the 1980s several authors presented realist interpretations which,
in retrospect, can be regarded as elaborations or variations on the
just-mentioned modal themes (for an overview and references, see Dieks
and Vermaas 1998). In spite of the differences among them, all the
modal interpretations agree on the following points:

The interpretation should be based on the standard formalism of
quantum mechanics, with one exception: the projection postulate is
left out.

The interpretation should be “realist” in the precise
sense that it assumes that quantum systems always possess a number of
definite properties, which may change with time. It should be noted,
however, that this semantic realism is compatible with agnosticism or
van Fraassen’s brand of empiricism (van Fraassen 1991, Bueno
2014), and does not presuppose epistemological realism.

Quantum mechanics is taken to be fundamental: it applies both to
microscopic and macroscopic systems.

The dynamical state of the system (pure or mixed) tells us what
the possible properties of the system and their corresponding
probabilities are. This is achieved by a precise mathematical rule
that specifies a probabilistic relationship between the dynamical
state and possible value states.

A quantum measurement is an ordinary physical interaction. There
is no collapse of the dynamical state (the wavefunction): the
dynamical state always evolves unitarily according to the
Schrödinger equation.

The Kochen-Specker theorem (1967) is a barrier to any realist
classical-like interpretation of quantum mechanics, since it proves
the impossibility of ascribing precise values to all physical
quantities (observables) of a quantum system simultaneously, while
preserving the functional relations between commuting observables.
(See the entry on the
the Kochen-Specker theorem.)
Therefore, realist non-collapse interpretations are committed to
selecting a privileged set of definite-valued observables out of all
observables. Each modal interpretation thus supplies a “rule of
definite-value ascription” or “actualization rule”,
which picks out, from the set of all observables of a quantum system,
the subset of definite-valued properties.

The question is: what should this actualization rule look like? Since
the mid-1990s a series of approaches faced this question (Clifton
1995a,b; Dickson 1995a,b; Dieks 1995). Each one of them proposed a
group of conditions that the set of definite-valued properties should
obey, and characterized this set in terms of the dynamical state
\(\ket{\phi}\) of the system. The common result was that the possible
value states of the components of a two-part composite system are
given by the states occurring in the Schmidt (bi-orthogonal)
decomposition of the dynamical state, or, equivalently, by the
projectors occurring in the spectral decomposition of the density
matrices representing partial systems (obtained by partial
tracing)—see Section 4 for more details.

The definite-valued properties have also been characterized somewhat
differently (Bub and Clifton 1996; for an improved version, see Bub,
Clifton and Goldstein 2000), that is, in terms of the quantum state
\(\ket{\phi}\) plus a “privileged observable”
\(\boldsymbol{R}\), which is privileged in the sense that it
represents a property that is always definite-valued (see also Dieks
2005, 2007). On this basis, Bub (1992, 1994, 1997) suggests that with
hindsight a number of traditional interpretations of quantum theory
can be characterized as modal interpretations. Notable among them are
the Dirac-von Neumann interpretation, (what Bub takes to be)
Bohr’s interpretation, and Bohm’s theory. Bohm’s
theory is a modal interpretation in which the privileged observable
\(\boldsymbol{R}\) is the position observable.

3. Atomic modal interpretation

The Hilbert space of the universe \(\mathcal{H}^{\univ}\),
like any Hilbert space, can be factorized in countless ways. If one
supposes that each factorization defines a legitimate set of
subsystems of the universe, the multiple factorizability implies that
there exists a multiplicity of ways of defining the building blocks of
nature. If the properties (value states) of all these quantum systems
are defined by means of the partial trace with respect to the rest of
the universe (see later for more details), it turns out that a
contradiction of the Kochen-Specker type arises (Bacciagaluppi
1995).

The Atomic Modal Interpretation (AMI, Bacciagaluppi and Dickson 1999)
tries to overcome this obstacle by assuming that there is in nature a
fixed set of mutually disjoint atomic quantum systems
\(S^j\) that constitute the building blocks of
all the other quantum systems. From the mathematical point of view,
this means that the Hilbert space \(\mathcal{H}^{\univ}\) of
the entire universe can only be meaningfully factorized in a single
way, which defines a preferred factorization. If each atomic
quantum system \(S^j\) is represented by its
corresponding Hilbert space \(\mathcal{H}^j\), then the
Hilbert space \(\mathcal{H}^{\univ}\) of the universe must be
written as

The main appeal of this idea is that it is in consonance with the
standard model of particle physics, where the fundamental blocks of
nature are the elemental particles, e.g., quarks, electrons, photons,
etc., and their interactions. The property ascription to the atomic
quantum systems in the AMI further follows the general idea of modal
interpretations, that is, the ascription depends via a fixed rule on
the dynamical state of the system.

The main challenge for the AMI is to justify the assumption that there
is a preferred partition of the universe and to provide some idea
about what this factorization should look like. AMI also faces a
conceptual problem. In this interpretation, a non-atomic quantum
system \(S^{\sigma}\), defined as composite of atomic quantum systems,
does not necessarily have properties that correspond to the outcomes
of measurements. The reason is that the system \(S^{\sigma}\) might be
in the quantum state \(\varrho^{\sigma}\) with an eigenprojector
\(\Pi^{\sigma}\) such that
\(\mathrm{Tr}^{\sigma}(\varrho^{\sigma}\Pi^{\sigma}) = 1\). This implies that
if one measured the property represented by \(\Pi^{\sigma}\), one
would obtain a positive outcome with probability 1. But it may be the
case that the projector \(\Pi^{\sigma}\) is not a composite of
atomic properties and, therefore, according to the AMI, it is not a
property possessed by the composite quantum system
\(S^{\sigma}\).

Two answers to this conceptual difficulty have been proposed. The
first allows the existence of dispositional properties in addition to
ordinary properties (Clifton 1996). According to the second answer,
the projector \(\Pi^{\sigma}\) of the composite system
\(S^{\sigma}\) shows that \(S^{\sigma}\) has a
collective dynamical effect onto the measurement device, that is, an
effect that cannot be explained by the action of the atomic components
(Dieks 1998). In other words, the composite quantum system, when
interacting with its environment, can behave as a collective entity,
screening off the contribution of the atomic quantum systems. This
means that sometimes a non-atomic quantum system
\(S^{\sigma}\) may be taken as if it were an atomic
quantum system within the framework of a coarse-grained
description.

In the biorthogonal-decomposition interpretation (BDMI, sometimes
known as “Kochen-Dieks modal interpretation”, Kochen 1985;
Dieks 1988, 1989a,b, 1994a,b), the definite-valued observables are
picked out by the biorthogonal (Schmidt) decomposition of the pure
quantum state of the system:

Biorthogonal Decomposition Theorem:
Given a vector \(\ket{\psi}\) in a tensor-product Hilbert space \(\mathcal{H}^1
\otimes \mathcal{H}^2\), there exist bases \(\{\ket{a_i}\}\) and \(\{\ket{p_i}\}\)
for \(\mathcal{H}^1\) and \(\mathcal{H}^2\) respectively, such that \(\ket{\psi}\) can be
written as a linear combination of terms of the form \(\ket{a_i}
\otimes \ket{p_i}\). If the absolute values (modulus) of the
coefficients in this linear combination are all unequal, then the
bases are unique (see, for example, Schrödinger 1935 for a
proof).

In quantum mechanics the theorem means that, given a composite system
consisting of two subsystems, its state picks out (in many cases,
uniquely) a basis for each of the subsystems. According to the BDMI,
those bases generate the definite-valued properties (the value states)
of the corresponding subsystems.

The BDMI is particularly appropriate to account for quantum
measurement. Let us consider an ideal measurement under the standard
von Neumann model, according to which a quantum measurement is an
interaction between a system \(S\) and a measuring apparatus
\(M\). Before the interaction, \(M\) is prepared in a
ready-to-measure state \(\ket{p_0}\), eigenvector of
the pointer observable \(P\) of \(M\), and the state of
\(S\) is a superposition of the eigenstates
\(\ket{a_i}\) of an observable \(A\) of
\(S\). The interaction introduces a correlation between the
eigenstates \(\ket{a_i}\) of \(A\) and the
eigenstates \(\ket{p_i}\) of \(P\):

In this case, according to the BDMI prescription, the preferred
context of the measured system \(S\) is defined by the set
\(\{\ket{a_i}\}\) and the preferred context of the measuring apparatus
\(M\) is defined by the set \(\{\ket{p_i}\}\). Therefore, the pointer
position is a definite-valued property of the apparatus: it acquires
one of its possible values (eigenvalues) \(p_i\). And analogously in
the measured system: the measured observable is a definite-valued
property of the measured system, and it acquires one of its possible
values (eigenvalues) \(a_i\).

In spite of the fact that this modal interpretation is characterized
by the central role played by biorthogonal decomposition, two
different versions can be distinguished. One of them adopts a
metaphysics in which all properties are relational and, as a
consequence, the fact that the application of the interpretation is
restricted to subsystems of a two-component compound system is not a
problem (Kochen 1985). This relation has been called
“witnessing”: properties are not possessed by the system
absolutely, but only when it is “witnessed” by another
system. Consider the measurement described above: the pointer
“witnesses” the value acquired by the measured observable
of the measured system.

By contrast, according to the other version (Dieks 1988, 1989a,b) the
properties ascribed to the system do not have a relational character.
This proposal therefore faces consistency questions about the
assignments of definite values to observables according to different
ways of splitting up the total system into components. Consider, for
example, the three-component composite system \(\alpha \beta \chi\). We
could apply the biorthogonal decomposition theorem to the
two-component system (i) \(\alpha(\beta \chi)\), or (ii)
\(\beta(\chi \alpha)\) or (iii) \(\chi(\alpha \beta)\). Suppose that, as a
result of this, in case (i) the system \(\alpha\) has the definite-valued
property \(P\), in case (ii) the system \(\beta\) has the
definite-valued property \(Q\), and in case (iii) the system
\(\alpha \beta\) has the definite-valued property \(R\). How do the
definite-valued properties of \(\alpha\) and \(\beta\) relate to those of
\(\alpha \beta\)? Are the definite-values properties of system
\(\alpha \beta\) \(P \amp Q\), or \(R\), or both?

This problem has been addressed by different authors during the 1990s
(see Vermaas 1999; Bacciagaluppi 1996). This work led to the
spectral-decomposition modal interpretation (SDMI, sometimes known as
“Vermaas-Dieks modal interpretation”, Vermaas and Dieks
1995) a generalization of the BDMI interpretation to mixed states. The
SDMI is based on the spectral decomposition of the reduced density
operator: the definite-valued properties \(\Pi_i\) of a system and
their corresponding probabilities \(\mathrm{Pr}_i\) are given by the non-zero
diagonal elements of the spectral decomposition of the system’s
state,

This new proposal matches the old one in cases where the old one
applies, and generalizes it by fixing the definite-valued properties
in terms of multi-dimensional projectors when the biorthogonal
decomposition is degenerate: definite-valued properties need not
always be represented by one-dimensional
vectors—higher-dimensional subspaces of the Hilbert space can
also occur.

The SDMI also has a direct application to the measurement situation.
Consider quantum measurement as described above, where the reduced
states of the measured system \(S\) and the measuring apparatus
\(M\) are

According to the SDMI, the preferred context of \(S\) is defined by
the projectors \(\Pi_i^a\) and the preferred context of \(M\) is
defined by projectors \(\Pi_i^p\). Therefore, also in the SDMI, the
observables \(A\) of \(S\) and \(P\) of \(M\) acquire actual definite
values, whose probabilities are given by the diagonal elements of the
diagonalized reduced states.

The SDMI faces the same difficulty as the non-relational version of
the BDMI: the fact that a system can be decomposed in a variety of
different ways. In particular, the factorization of a given Hilbert
space \(\mathcal{H}\) into two factors, \(\mathcal{H} = \mathcal{H}^1 \otimes \mathcal{H}^2\), can be
“rotated” to produce different factorizations \(\mathcal{H}' =
\mathcal{H}^1{}^\prime \otimes \mathcal{H}^2{}^\prime\). Are we to apply the SDMI to each
such factorization? How are the results related, if at all? A theorem
due to Bacciagaluppi (1995, see also Vermaas 1997) shows, in essence,
that if one applies the SDMI to the “subsystems” obtained
in every factorization and insists that the definite-valued properties
so-obtained are not relational, then one will be led to a mathematical
contradiction of the Kochen-Specker variety. In response, one could
adopt the view that subsystems have their definite-valued properties
“relative to a factorization”; we will come back to this
issue below.

Healey (1989) was also among the first to make use of the biorthogonal
decomposition theorem, developing these ideas in a somewhat different
direction. His main concern was the apparent non-locality of quantum
mechanics. Healey’s intuition about the way a modal
interpretation based on the biorthogonal decomposition theorem would
be applied to, say, an EPR experiment is to implement the idea that an
EPR pair possesses a “holistic” property; this can then
explain why the apparatus on one side of the experiment acquires a
property that is correlated to the result on the other side.

In Healey’s proposal, the biorthogonal decomposition theorem is
used, but the set of possible properties is subsequently modified in
order to fulfill a variety of desiderata. The first is consistency:
the aim is to avoid Kochen-Specker-type results. A second is to
maintain a plausible theory of the relationship between composite
systems and their subsystems. A third is to maintain a plausible
account of the relations among definite-valued properties at a given
time. A fourth is to maintain a plausible account of the relations
among definite-valued properties at different times. The structure of
definite-valued properties that emerges from these conditions is
extremely complicated. Some progress has been made since
Healey’s book was published (see for example Reeder and Clifton
1995) but, in general, it remains difficult to see what the set of
definite-valued properties is according to his approach.

5. Non-ideal measurements

Above we suggested that the BDMI and the SDMI solve the measurement
problem in a particularly direct way. This is right in the case of the
ideal von Neumann measurement, as explained in the previous section,
where the eigenstates \(\ket{a_i}\) of an observable
\(A\) of the measured system \(S\) are perfectly correlated
with the eigenstates \(\ket{p_i}\) of the pointer
\(P\) of the measuring apparatus \(M\). However, ideal
measurement is a situation that can never be achieved in practice: the
interaction between \(S\) and \(M\) never introduces a
completely perfect correlation. Two kinds of non-ideal measurements
are usually distinguished in the literature:

Note, however, that disturbing measurement can be rewritten as
imperfect measurements (and vice versa).

Imperfect measurements pose a challenge to the BDMI and the SDMI,
since their rules for selecting the definite-valued properties do not
pick out the right properties for the apparatus in the imperfect case
(see Albert and Loewer 1990, 1991, 1993; also Ruetsche 1995). An
example that clearly brings out the difficulties introduced by
non-ideal measurements was formulated in the context of Stern-Gerlach
experiments (Elby 1993). This argument uses the fact that the
wavefunctions in the \(z\)-variable typically have infinite
“tails” that introduce non-zero cross-terms; therefore,
the “tail” of the wavefunction of the “down”
beam may produce detection in the upper detector, and vice versa (see
Dickson 1994 for a detailed discussion).

In fact, if the biorthogonal decomposition is applied to the
non-perfectly correlated state \(\sum_{ij} d_{ij} \ket{a_i} \otimes
\ket{p_j} = \sum_i c_i' \ket{a_i'} \otimes \ket{p_i'}\), according to
the BDMI the result does not select the pointer \(P\) as a
definite-valued property, but a different observable \(P'\) with
eigenstates \(\ket{p_i'}\). In this case, in which the definite-valued
properties selected by a modal interpretation are different from those
expected, the question arises how different they are. In the case of
an imperfect measurement, it may be assumed that the \(d_{ij} \ne 0\),
with \(i \ne j\), be small; then, the difference might be also
small. But in the case of a disturbing measurement, the \(d_{ij} \ne
0\), with \(i \ne j\), need not be small and, as a consequence, the
disagreement between the modal interpretation assignment and the
experimental result might be unacceptable (see a full discussion in
Bacciagaluppi and Hemmo 1996). This fact has been considered as a
“silver bullet” for killing the modal interpretations
(Harvey Brown, cited in Bacciagaluppi and Hemmo 1996).

There is another important problem related to non-ideal measurements.
When the final state of the composite system (measured system plus
measuring device) is very nearly degenerate when written in the basis
given by the measured observable and the apparatus’s pointer
(that is, when the probabilities for the various results are nearly
equal), the spectral decomposition does not, in general, select as
definite-valued properties close to those ideally expected. In fact,
the observables so selected may be incompatible (non-commuting) with
the observables that we expect on the basis of observation
(Bacciagaluppi and Hemmo 1994, 1996).

In order to face the problems that non-ideal measurements pose to the
BDMI and the SDMI, several authors have appealed to the phenomenon of
decoherence; this will be discussed below.

6. Properties of composite systems

Let us take a composite system \(\alpha \beta\), whose component
subsystems \(\alpha\) and \(\beta\) are represented by the Hilbert
spaces \(\mathcal{H}^{\alpha}\) and \(\mathcal{H}^{\beta}\), respectively, and consider a
property represented by the projector \(\Pi^{\alpha}\) defined on
\(\mathcal{H}^{\alpha}\). It is usual to assume that \(\Pi^{\alpha}\) represents
the same property as that represented by \(\Pi^{\alpha} \otimes
I^{\beta}\) defined on \(\mathcal{H}^{\alpha} \otimes \mathcal{H}^{\beta}\), where
\(I^{\beta}\) is the identity operator on \(\mathcal{H}^{\beta}\). This
assumption is based on the observational indistinguishability of the
magnitudes represented by \(\Pi^{\alpha}\) and \(\Pi^{\alpha} \otimes
I^{\beta}\): if the \(\Pi^{\alpha}\)-measurement has a certain
outcome, then the \(\Pi^{\alpha} \otimes I^{\beta}\)-measurement has
exactly the same outcome.

The question is then: If the rules of the BDMI and the SDMI applied to
\(\alpha\) assign a value to \(\Pi^{\alpha}\), do those rules applied
to the composite system \(\alpha \beta\) assign the same value to
\(\Pi^{\alpha} \otimes I^{\beta}\) (condition known as Property
Composition), and vice versa (Property Decomposition)? The answer to
this question is negative: the BDMI and the SDMI violate Property
Composition and Property Decomposition (for a proof, see Vermaas
1998).

Of course, if one maintains that the projectors \(\Pi^{\alpha}\) and
\(\Pi^{\alpha} \otimes I^{\beta}\) represent the same property, the
violation of Property Composition and Property Decomposition is a
serious problem for any interpretation. This is the position adopted
by Arntzenius (1990), who judges this violation to be bizarre, since
it assigns different truth values to propositions like ‘the
left-hand side of a table is green’ and ‘the table has a
green left-hand side’, which are normally not distinguished; a
similar argument is put forward by Clifton (1996, see also Clifton
1995c).

However, Vermaas (1998) argues that the observational
indistinguishability of the magnitudes represented by \(\Pi^{\alpha}\)
and \(\Pi^{\alpha} \otimes I^{\beta}\) does not force one to consider
these two projectors as representing the same property: in fact, they
are distinguishable from a theoretical viewpoint, since they are
defined on different Hilbert spaces. Moreover, he argues that the
examples developed by Arntzenius and Clifton sound bizarre precisely
in the light of Property Composition and Property Decomposition. But
in the quantum realm we must accept that the questions of which
properties are possessed by a system and which by its subsystems are
different questions: the properties of a composite system \(\alpha
\beta\) don’t reveal information about the properties of
subsystem \(\alpha\), and vice versa. Vermaas concludes that the tenet
that \(\Pi^{\alpha}\) and \(\Pi^{\alpha} \otimes I^{\beta}\) do
represent the same property can be viewed as an addition to quantum
mechanics, which can be denied as, for instance, van Fraassen (1991)
did.

7. Dynamics of properties

As we have seen, modal interpretations intend to provide, for every
instant, a set of definite-valued properties and their probabilities.
Some advocates of modal interpretations may be willing to leave the
matter, more or less, at that. Others take it to be crucial for any
modal interpretation that it also answers questions of the form: Given
that the property \(P\) of a system has the actual value \(\alpha\) at
time \(t_0\), what is the probability that its property \(P'\) has the
actual value \(\beta\) at time \(t_1 \gt t_0\)? In other words, they
want a dynamics of actual properties.

There are arguments on both sides. Those who argue for the necessity
of such a dynamics maintain that we have to assure that the
trajectories of actual properties really are, at
least for macroscopic objects, like we see them to be, i.e., like the
records contained in memories. For example, we should require not only
that the book at rest on the desk possess a definite location, but
also that, if undisturbed, its location relative to the desk does not
change in time. Accordingly, one cannot get away with simply
specifying the definite properties at each instant of time. We need
also to show that this specification is at least compatible with a
reasonable dynamics; better still, specify this dynamics
explicitly.

Those who consider a dynamics of actual properties to be superfluous
reply that such a dynamics is more than what an interpretation of
quantum mechanics needs to provide. Memory contents for each instant
are enough to make empirical adequacy possible.

As pointed out by Ruetsche (2003), in this debate about the need for a
dynamics of actual properties it is important whether the modal
interpretation is viewed as leading to a hidden-variables
theory, in which value states are added as hidden variables to
the original formalism in order to obtain a full description of the
physical situation, or rather as only equipping the original formalism
with a new semantics. In the first approach one would expect a full
dynamics of actual properties, in the second this is not so clear.

Of course, modal interpretations do admit a trivial dynamics, namely,
one in which there is no correlation from one time to the next. In
this case, the probability of a transition from the property \(P\)
having the actual value \(\alpha\) at \(t_0\), to the property \(P'\)
having the actual value \(\beta\) at \(t_1 \gt t_0\) is just the
single-time probability for \(P'\) having \(\beta\) at
\(t_1\). However, this dynamics is unlikely to interest those who feel
the need for a dynamics at all. Several researchers have contributed
to the project of constructing a more interesting form of dynamics for
modal interpretations (see Vermaas 1996, 1998). An important account
is due to Bacciagaluppi and Dickson (1999, see also Bacciagaluppi
1998). That work shows the most significant challenges that the
construction of a dynamics of actual properties must face.

The first challenge is posed by the fact that the set of
definite-valued properties—let us call it
‘\(S\)’—may change over time. One therefore has to
define a family of maps, each one being a 1–1 map from \(S_0\)
at time \(t_0\) to a different \(S_t\) at time \(t\), for any
time. With such a family of maps, one can effectively define
conditional probabilities within a single state space, and then
translate them into “transition” probabilities. For this
technique to work, \(S_t\) must have the same cardinality at any time.
However, in general this is not the case: for instance, in the SDMI,
the number of different projectors appearing in the spectral
decomposition of the density matrix may vary with time.

A way out of this is to augment \(S\) at each time so that its
cardinality matches the highest cardinality that \(S\) ever
achieves. Of course, one hopes to do so in a way that is not
completely ad hoc. For example, in the context of the SDMI,
Bacciagaluppi, Donald and Vermaas (1995) show that the
“trajectory” through Hilbert space of the spectral
components of the reduced state of a physical system will, under
reasonable conditions, be continuous, or have only isolated
discontinuities, so that the trajectory can be naturally extended to a
continuous trajectory (see also Donald 1998). This result suggests a
natural family of maps as discussed above: map each spectral component
at one time to its unique continuous evolved component at later
times.

The second challenge to the construction of a dynamics arises from the
fact that one wants to define transition probabilities over
infinitesimal units of time, and then derive the finite-time
transition probabilities from them. Bacciagaluppi and Dickson (1999)
argue that, adapting results from the theory of stochastic processes,
one can show that the procedure may, more or less, be carried out for
modal interpretations of at least some varieties.

Finally, one must actually define infinitesimal transition
probabilities that will give rise to the proper quantum-mechanical
probabilities at each time. Following earlier papers by Bell (1984),
Vink (1993) and others, Bacciagaluppi and Dickson (1999) define an
infinite class of such infinitesimal transition probabilities, such
that all of them generate the correct single-time probabilities, which
arguably are all we can really test. However, Sudbery (2002) has
contended that the form of the transition probabilities would be
relevant to the precise form of spontaneous decay or the
“Dehmelt quantum jumps”; he independently developed the
dynamics of Bacciagaluppi and Dickson and applied it in such a way
that it leads to the correct predictions for these experiments.
Gambetta and Wiseman (2003, 2004) developed a dynamical modal account
in the form of a non-Markovian process with noise, also extending
their approach to positive operator-valued measures (POVMs). More
recently, Hollowood (2013a, 2013b, 2014) has elaborated the idea that
the dynamics of value states can be modeled by a discrete-time Markov
chain.

8. Perspectival modal interpretation

As we have seen, both the SDMI and the non-relational version of the
BDMI have to face the problem of the multiple factorizability of a
given Hilbert space: if the definite-valued properties are monadic
(i.e., non-relational), both interpretations led to a
Kochen-Specker-type contradiction (Bacciagaluppi 1995). This points to
the direction of an interpretation that makes properties relational,
in this case relative to a factorization.

Extending this idea, a perspectival modal interpretation (PMI, Bene
and Dieks 2002) was developed, in which the properties of a physical
system have a relational character and are defined with respect to
another physical system that serves as a “reference
system” (see Bene 1997). This interpretation is similar in
spirit to the idea that systems have properties as
“witnessed” by the rest of the universe (Kochen 1985).
However, the PMI goes further by defining states of a system not only
with respect to the universe, but also with respect to arbitrary
larger systems. The PMI is closely related to the SDMI since similar
rules are used to assign properties to quantum systems.

In the PMI, the state of any system \(S\) needs the specification of a
“reference system” \(R\) with respect to which the state
is defined: this state of \(S\) with respect to \(R\) is denoted by
\(\varrho_{R}^{S}\). In the special case in which \(R\) coincides with
\(S\), the state \(\varrho_{S}^{S}\) is called “the state of
S with respect to itself”. If the system \(S\) is contained
in a system \(A\), the state \(\varrho_{A}^{S}\) is defined as the
density operator that can be derived from \(\varrho_{A}^{A}\) by
taking the partial trace over the degrees of freedom in \(A\) that do
not pertain to \(S\):

\[
\varrho_{A}^{S} = \mathrm{Tr}_{(A\setminus S)} \varrho_{A}^{A}
\]

With these definitions, the point of departure of the PMI is the
quantum state of the whole universe with respect to itself, which it
is assumed to be a pure state \(\varrho_{U}^{U} = \ket{\psi}
\bra{\psi}\) which evolves unitarily according to the Schrödinger
equation. For any system \(S\) contained in the universe, its state
with respect to itself \(\varrho_{S}^{S}\) is postulated to be one of
the projectors of the spectral resolution of

In particular, if there is no degeneracy among the eigenvalues of
\(\varrho_{U}^{S}\), these projectors are one-dimensional and
\(\varrho_{S}^{S}\) is the one-dimensional projector
\(\ket{\psi_{S}} \bra{\psi_{S}}\).

Within this PMI conceptual framework it can be shown that a system may
be localized from the perspective of one observer and, nevertheless,
may be delocalized from a different perspective. But it also follows
that observers who look at the same macroscopic object, at the same
time and under identical circumstances, will see it (practically) at
the same spot.

The core idea of this interpretation is that all different relational
descriptions, given from different perspectives, are equally objective
and all correspond to physical reality (which has a relational
character itself). We cannot explain the relational states by
appealing to a definition in terms of more basic non-relational
states. Further analysis shows that in this interpretation EPR-type
situations can be understood in a basically local manner. Indeed, the
change in the relational state of particle 2 with respect to the
2-particle system can be understood as a consequence of the change in
the reference system brought about by the local measurement
interaction between particle 1 and the measuring device. This local
measurement is responsible for the creation of a new perspective, and
from this new perspective there is a new relational state of particle
2 (see also Dieks 2009).

The PMI agrees with Bohr’s qualitative argument that any
reasonable definition of physical reality in the quantum realm should
include the experimental setup. However, the PMI is more general in
the sense that the state of a system is defined with respect to any
larger physical system, not necessarily an instrument. This removes
the threat of subjectivism, since the relational states follow
unambiguously from the quantum formalism and the physics of the
situation.

It is interesting to consider the connections between the PMI and
other relational proposals. For instance, Berkovitz and Hemmo (2006)
propose the prospects of a relational modal interpretation in the
relativistic case (we will come back to this point below). In turn,
Rovelli and coworkers propose an explicit ‘relational quantum
mechanics’ that emphasizes the possibility of different
descriptions of a physical system depending on the perspective
(Rovelli 1996; Rovelli and Smerlak 2007; Laudisa and Rovelli 2008; see
also van Fraassen 2010 and the entry on
relational quantum mechanics).
In spite of the points of contact between the PMI and Rovelli’s
relational interpretation, there are significant differences. In
Rovelli’s proposal, the concepts of measurement interaction and
of definite outcomes of measurements are primary; moreover, the state
has to be updated every time that a measurement event occurs and, as a
consequence, it changes discontinuously with every new event. On the
contrary, the PMI is a realist interpretation where a measurement is
nothing else than a quantum interaction, and where unitary evolution
is the main dynamical principle, also when systems interact (see Dieks
2009).

9. Modal-Hamiltonian interpretation

As Bub (1997) points out, in most modal interpretations the preferred
context of definite-valued observables depends on the state of the
system. An exception is Bohmian mechanics, in which the preferred
context is a priori defined by the position observable; in this case,
property composition and property decomposition hold. But this is not
the only reasonable possibility for a modal interpretation with a
fixed preferred observable. In fact, the modal-Hamiltonian
interpretation (MHI, Lombardi and Castagnino 2008; Ardenghi,
Castagnino, and Lombardi 2009; Lombardi, Castagnino, and Ardenghi
2010; Ardenghi and Lombardi 2011) endows the Hamiltonian of a system
with a determining role, both in the definition of systems and
subsystems and in the selection of the preferred context.

The MHI is based on the following postulates:

Systems postulate (SP):
A quantum system \(S\) is represented by a pair \((\mathcal{O}, H)\)
such that (i) \(\mathcal{O}\) is a space of self-adjoint operators on a Hilbert
space, representing the observables of the system, (ii) \(H \in \mathcal{O}\) is
the time-independent Hamiltonian of the system \(S\), and (iii) if
\(\varrho_0 \in \mathcal{O}'\) (where \(\mathcal{O}'\) is the dual space of \(\mathcal{O})\) is the
initial state of \(S\), it evolves according to the Schrödinger
equation.

Although any quantum system can be decomposed in parts in many ways,
according to the MHI a decomposition leads to parts which are also
quantum systems only when the components’ behaviors are dynamically
independent of each other, that is, when there is no interaction among
the subsystems:

Composite systems postulate (CSP):
A quantum system represented by \(S: (\mathcal{O}, H)\), with initial state
\(\varrho_0 \in \mathcal{O}'\), is composite when it can be partitioned
into two quantum systems \(S^1 : (\mathcal{O}^1, H^1)\) and \(S^2 : (\mathcal{O}^2, H^2)\)
such that (i) \(\mathcal{O} = \mathcal{O}^1 \otimes \mathcal{O}^2\), and (ii) \(H = H^1 \otimes I^2 + I^1 \otimes H^2\) (where \(I^1\) and \(I^2\) are the identity
operators in the corresponding tensor product spaces). In this case,
we say that \(S^1\) and \(S^2\) are subsystems of the
composite system \(S = S^1 \cup S^2\). If the system is not composite,
it is elemental.

With respect to the preferred context, the basic idea of the MHI is
that the Hamiltonian of the system defines actualization. Any
observable that does not have the symmetries of the Hamiltonian cannot
acquire a definite actual value, since this actualization would break
the symmetry of the system in an arbitrary way.

Actualization rule (AR):
Given an elemental quantum system represented by \(S: (\mathcal{O}, H)\), the
actual-valued observables of \(S\) are \(H\) and all the observables
commuting with \(H\) and having, at least, the same symmetries as
\(H\).

The selection of the preferred context exclusively on the basis of a
preferred observable has been criticized by arguing that in the
Hilbert space formalism all observables are on an equal footing.
However, quantum mechanics is not just Hilbert space mathematics: it
is a physical theory that includes a dynamical law in which the
Hamiltonian is singled out to play a central role.

The justification for selecting the Hamiltonian as the preferred
observable ultimately lies in the success of the MHI and its ability
to solve interpretive difficulties. With respect to the first point:
the scheme has been applied to several well-known physical situations
(free particle with spin, harmonic oscillator, free hydrogen atom,
Zeeman effect, fine structure, the Born-Oppenheimer approximation),
leading to results consistent with empirical evidence (Lombardi and
Castagnino 2008, Section 5). With respect to interpretation, the MHI
confronts quantum contextuality by selecting a preferred context, and
has proved to be able to supply an account of the measurement problem,
both in its ideal and its non-ideal versions; moreover, in the
non-ideal case it gives a criterion to distinguish between reliable
and non-reliable measurements (Lombardi and Castagnino 2008, Section
6), a criterion that can be generalized when expressed in
informational terms (Lombardi, Fortin and López 2015).

In the MHI property composition and property decomposition hold
because the actualization rule only applies to elemental
systems: the definite-valued properties of composite systems are
selected on the basis of those of the elemental components, and
following the usual quantum assumption according to which the
observable \(A^1\) of a subsystem \(S^1\) and the observable \(A = A^1
\otimes I^2\) of the composite system \(S = S^1 \cup S^2\) represent
the same property (Ardenghi and Lombardi 2011).

The preferred context of the MHI does not change with time: the
definite-valued observables always commute with the Hamiltonian and,
therefore, they are constants of motion of the system. This means that
they are the same during the whole “life” of the quantum
system as a closed system, since its initial “birth”, when
it arises as a result of an interaction, up to its final
“death”, when it disappears by interacting with another
system. As a consequence, there is no need of accounting for the
dynamics of the actual properties as in the BDMI and the SDMI.

In more recent years, the MHI has extended its applications to further
situations, such as the non-collapse account of consecutive
measurements in physics (Ardenghi, Lombardi and Narvaja 2013) and the
problem of optical isomerism in chemistry (Fortin, Lombardi and
Martínez González 2016a, 2016b). Moreover, on the basis
of its closed-system perspective, the MHI opens the way toward a
top-down view of quantum mechanics, according to which reduced states
are coarse-grained states of a closed system (Fortin and Lombardi
2014) and decoherence is a phenomenon relative to the particular
partition of the closed system considered in each case (Lombardi,
Fortin and Castagnino 2012, Fortin and Lombardi 2016).

10. The interpretation of probability

One of the leading ideas of the modal interpretations is
probabilism: quantum mechanics does not correspond in a
one-to-one way to actual reality, but rather provides us with a list
of possibilities and their probabilities. Therefore, the notions of
possibility and probability are central in this interpretive
framework. This raises two issues: the formal treatment of
probabilities, and the interpretation of probability.

Since the set of events corresponding to all projector operators on a
given Hilbert space does not have a Boolean structure, the Born
probability (which is defined over these projectors) does not satisfy
the definition of probability of Kolmogorov (which applies to a
Boolean algebra of events). For this reason, some authors define a
generalized non-Kolmogorovian probability function over the
ortho-algebra of quantum events (Hughes 1989; Cohen 1989). Modal
interpretations do not follow this path: they conceive probabilities
as represented by a Kolmogorovian measure on the Boolean algebra
representing the definite-valued quantities, generated by mutually
commuting projectors. The various modal interpretations differ from
each other in their definitions of the preferred context on which the
Kolmogorovian probability is defined.

As we have seen, the definite-valued properties of a system are
usually characterized in terms of the quantum state \(\ket{\phi}\) and
a privileged observable \(\boldsymbol{R}\) (Bub and Clifton 1996; Bub,
Clifton, and Goldstein 2000; Dieks 2005). Dieks (2007) derives a
uniqueness result, namely that given the splitting of a total Hilbert
space into two factors spaces, representing the system and its
environment, respectively, the Boolean lattice of definite-valued
observables is fixed by the state of the system alone. Furthermore,
it follows that the Born measure is the only one that is definable
from just the product structure of Hilbert space, the state in the
Hilbert space, and the definite-valued observables selected by the
state.

The MHI defines a context as a complete set of orthogonal projectors
\(\{\Pi_{\alpha}\}\), such that \(\sum_{i} \Pi_{i} = I\) and
\(\Pi_{i}\Pi_j = \delta_{ij}\Pi_{i}\), where
\(I\) is the identity operator in \(\mathcal{H} \otimes \mathcal{H}\).
Since each context generates a Boolean structure, the state of the
system defines a Kolmogorovian probability function on each individual
context (Lombardi and Castagnino 2008). However, only the
probabilities defined on the context determined by the eigenprojectors
of the Hamiltonian of an elemental closed system correspond to the
possible values one of which becomes actual.

In modal interpretations the event space on which the (preferred)
probability measure is defined is a space of possible events,
among which only one becomes actual. The fact that the actual event is
not singled out by these interpretations is what makes them
fundamentally probabilistic. This aspect distinguishes modal
interpretations from many-worlds interpretations, where the
probability measure is defined on a space of events that are all
actual. Nevertheless, this does not mean that all modal
interpretations agree about the interpretation of probability.

In the context of the BDMI, the SDMI and the PMI, it is usually
claimed that, given the space of possible events, the state generates
an ignorance-interpretable probability measure over this set: quantum
probabilities quantify the ignorance of the observer about the actual
values acquired by the system’s observables (see, e.g., Dieks
1988; Clifton 1995a; Vermaas 1999; Bene and Dieks 2002).

By contrast to actualism—the conception that reduces possibility
to actuality (see Dieks 2010, Bueno 2014)—some modal
interpretations, in particular the MHI, adopt a possibilist
conception, according to which possible
events—possibilia—constitute a basic ontological
category (see Menzel 2007). The probability measure is in this case
seen as a representation of an ontological propensity of a possible
quantum event to become actual (Lombardi and Castagnino 2008; see also
Suárez 2004).

These views do not all exclude each other. If probabilities quantify
ignorance about the actual values of the observables, this need not
mean that this ignorance can be removed by the addition of further
information. If quantum probabilities are ontological propensities,
our ignorance about the possible event that becomes actual is a
necessary consequence of the indeterministic nature of the system
because there simply is no additional information specifying a more
accurate state of the system.

11. The role of decoherence

According to the environment-induced approach to decoherence (Zurek
1981, 2003; see also Schlosshauer 2007), the measuring apparatus is an
open system in continuous interaction with its environment; as a
consequence of this interaction, the reduced state of the apparatus
and the measured system becomes, almost instantaneously,
indistinguishable from a state that would represent an ignorance
mixture (“proper mixture”) over unknown values of the
apparatus’ pointer. The idea that decoherence might play a role in
modal interpretations was proposed by several authors early on (Dieks
1989b; Healey 1995). But it has acquired a central relevance in
relation to the discussion of non-ideal measurements in the modal
interpretation.

As we have seen, in the BDMI and the SDMI, the biorthogonal or the
spectral decomposition does not pick out the right properties for the
apparatus in non-ideal measurements. Bacciagaluppi and Hemmo (1996)
show that, when the apparatus is a finite-dimensional system in
interaction with an environment with a huge number of degrees of
freedom, decoherence guarantees that the spectral decomposition of the
apparatus’ reduced state will be very close to the ideally expected
result and, as a consequence, the apparatus’ pointer
is—approximately—selected as an actual definite-valued
observable. Alternatively, Bub (1997) proposes that it is not
decoherence—with the “tracing out” of the
environment and the diagonalization of the reduced state of the
apparatus—that is relevant for the definite value of the
pointer, but the triorthogonal or \(n\)-orthogonal decomposition
theorem, since it singles out a unique pointer basis for the
apparatus.

In either case, the interaction with the environment is a great help
to the BDMI and the SDMI for handling non-ideal measurements with
finite-dimensional apparatuses. However, the case of
infinitely many distinct states for the apparatus is more
troublesome. Bacciagaluppi (2000) has analyzed this situation, using a
continuous model of the apparatus’ interaction with the environment.
He concludes that in this case the spectral decomposition of the
reduced state of the apparatus generally does not pick out states that
are close enough to the ideally expected state. This result seems to
apply also to other cases where a macroscopic system (not described as
finite-dimensional) experiences decoherence due to interaction with
its environment (see Donald 1998). However, model calculations in
perspectival versions of the modal interpretation (Bene and Dieks
2002; Hollowood 2013a, 2013b, 2014) indicate that the problem is less
severe in realistic circumstances than originally supposed.

As said above, in the case of the MHI decoherence is not explicitly
appealed to in order to account for the definite reading of the
apparatus’ pointer (neither in ideal nor in non-ideal measurements).
However, there still is a relation with the decoherence program. In
fact, the measuring apparatus is always a macroscopic system with a
huge number of degrees of freedom, and the pointer must be a
“collective” and empirically accessible observable; as a
consequence, the many degrees of freedom corresponding to the
degeneracies of the pointer play the role of a decohering
“internal environment” (for details, see Lombardi 2010;
Lombardi et al. 2011). The role of decoherence in the MHI becomes
clearer when the phenomenon of decoherence is understood from a
closed-system perspective (Castagnino, Laura, and Lombardi 2007;
Castagnino, Fortin, and Lombardi 2010; Lombardi, Fortin, and
Castagnino 2012). (See the entry on
the role of decoherence in quantum mechanics.)

12. Open problems and perspectives

There are a number of open problems and perspectives in the modal
program. Here we will consider some of them.

Modal interpretations are based on the standard formalism of quantum
mechanics (in the Hilbert space version or in the algebraic version).
However, Brown, Suárez and Bacciagaluppi (1998) argue that
there is more to quantum reality than what is described by
operators and quantum states: they claim that gauges and coordinate
systems are important to our description of physical reality as well,
while modal interpretations (AM, BDMI and SDMI) have standardly not
taken such things into consideration. In a similar vein, it has been
argued that the Galilean space-time symmetries endow the formal
skeleton of quantum mechanics with the physical flesh and blood that
identify the fundamental physical magnitudes and that allow the theory
to be applied to concrete physical situations (Lombardi and Castagnino
2008). The set of definite-valued observables of a system should be
left invariant by the Galilean transformations: it would be
unacceptable that this set changed as a mere result of a change in the
perspective from which the system is described. On the basis of this
idea, the MHI rule of actualization has been reformulated in an
explicitly invariant form, in terms of the Casimir operators of the
Galilean group (Ardenghi, Castagnino, and Lombardi 2009; Lombardi,
Castagnino, and Ardenghi 2010).

Another fundamental question is the relativistic extension of the
modal approach. Dickson and Clifton (1998) have shown that a large
class of modal interpretations of ordinary quantum mechanics cannot be
made Lorentz-invariant in a straightforward way (see also Myrvold
2002). With respect to the extension to algebraic quantum field theory
(see Dieks 2002; Kitajima 2004), Clifton (2000) proposed a natural
generalization of the non-relativistic modal scheme, but Earman and
Ruetsche (2005) showed that it is not yet clear whether it will be
able to deal with measurement situations and whether it is empirically
adequate. The problems revealed by these investigations are due to the
non-relativistic nature of the formalism of quantum mechanics that is
employed, in particular to the fact that the concept of a state of an
extended system at one instant is central. In a local
field-theoretic context this becomes different, and this may avoid
conflicts with relativity (Earman and Ruetsche 2005). Berkovitz and
Hemmo (2005) and Hemmo and Berkovitz (2005) propose a different way
out: they argue that perspectivalism can come to the rescue here (see
also Berkovitz and Hemmo 2006). In turn, in the context of the MHI, it
has been argued that the actualization rule, expressed in terms of the
Casimir operators of the Galilean group in non-relativistic quantum
mechanics, can be transferred to the relativistic domain by changing
the symmetry group accordingly: the definite-valued observables of a
system would be those represented by the Casimir operators of the
Poincaré group. Since the mass operator and the squared spin
operator are the only Casimir operators of the Poincaré group,
they would always be definite-valued observables. This conclusion
would be in agreement with a usual assumption in quantum field theory:
elemental particles always have definite values of mass and spin, and
those values are precisely what define the different kinds of
elemental particles of the theory (Lombardi and Fortin 2015).

There are also specifically philosophical issues concerning
ontological matters: about the nature of the items referred to by
quantum mechanics, that is, about the basic categories of the quantum
ontology. As we have seen, in general the properties of quantum
systems are considered to be monadic, with the exception of the
relational version of the BDMI and the PMI where these properties are
relational. In any case, it might be asked whether a quantum system
has to be conceived as an individual substratum supporting properties
or as a mere “bundle” of properties. Following an original
idea of Lombardi and Castagnino (2008), da Costa, Lombardi and Lastiri
(2013) and da Costa and Lombardi (2014) have suggested that, in the
modal context, the bundle view might be appropriate to supply an
answer to the problem of indistinguishability (see also French and
Krause 2006). Nevertheless, this quantum ontology of propertied does
not prevent the emergence of particles under certain particular
circumstances (see Lombardi and Dieks 2016).

Recently, modal interpretations have begun to be considered by
practicing physicists and mathematicians interested in foundational
matters. For instance, Hollowood (2014) offers an interpretation of
quantum mechanics inspired by the perspectival modal interpretation:
the state of an open system describes its properties from the
perspective of the closed system of which it is a sub-system. In turn,
Barandes and Kagan (2014a, 2014b) propose a “minimal modal
interpretation”, inspired by the SDMI, according to which the
preferred context is given by the evolving reduced state of the open
system. Nakayama (2008a, 2008b) has explored connections between the
modal interpretation and the framework of topos theory.

These and similar developments have arisen in the context of detailed
technical investigations. This illustrates one of the advantages of
the modal approach: it makes use of a precise set of rules that
determine the set of definite-valued observables, and this makes it
possible to derive rigorous results. It may well be that several of
these results, e.g., no-go theorems, can be applied to other
interpretations as well (e.g., to the many-worlds interpretation, see
Dieks 2007). Whatever the merit of the modal ideas in the end, one can
at least say that they have given rise to a serious and fruitful
series of investigations into the nature of quantum theory.

Bub, J., and R. Clifton, 1996, “A uniqueness theorem for
interpretations of quantum mechanics,” Studies in History
and Philosophy of Modern Physics, 27: 181–219.

Bub, J., R. Clifton, and S. Goldstein, 2000, “Revised proof
of the uniqueness theorem for ‘no collapse’
interpretations of quantum mechanics,” Studies in History
and Philosophy of Modern Physics, 31: 95–98.

–––, 1994a, “Objectification, measurement
and classical limit according to the modal interpretation of quantum
mechanics,” in P. Busch, P. Lahti, and P. Mittelstaedt (eds.),
Proceedings of the Symposium on the Foundations of Modern
Physics, Singapore: World Scientific, pp. 160–167.

Acknowledgments

As of the December 2012 update, the credited authors for this entry
are Olimpia Lombardi and Dennis Dieks. The
original version of this entry
(published in 2002, last archived in Fall 2007) was authored solely
by Michael Dickson and we acknowledge that some sentences of that first version are still part of the current entry (particularly in Section 7).